Movies vs show.
Okay, this is a more even fight. Let's dive right in.
Size:
Smaug's size
Smaug was called "twice as big as a 747" by a VFX supervisor who worked on the films. Weta made a chart of him, showing that he's about 426-498 feet long.
Drogon's size
Drogon is approximately 128 feet long by my estimation. The director of the show says he's the size of a 747, which are about 160-250 feet long.
Speed:
Smaug's speed
Smaug has reached 600 mph in level flight. His "fast cruising" (air combat, probably) speed has been calced by Screen Junkies at 260 mph.
Drogon's speed
Screen Junkies calced Drogon's cruising speed at 160 mph in the same video. From that I think we can cautiously extrapolate a midair combat speed just over 200 mph.
Agility:
Smaug's agility
Smaug does not appear to be very agile—he never performs any very difficult maneuvers mid-air.
Drogon's agility
...but neither does the winged shadow. I'll give a slight edge to Drogon due to his smaller size.
Armor:
Smaug's armor
Smaug's armor requires at least 37.548 GPa to be penetrated by an arrow.
As for heat tolerance, he flies through his own fire several times, no-sold blue-hot explosions, and shrugged off being submerged in molten gold, which has a temperature equal to some lava. Note that the gold touched his weak spot, so his bare skin has a very high temperature tolerance (molten gold sticks to objects and thus would transfer a very large amount of heat). Also, his jaws appeared to be open when he went under, which indicates a very high temperature tolerance even inside his mouth, and there isn't any gold in his mouth when he comes back out, so presumably he swallowed it.
Drogon's armor
Drogon was injured by a ballista bolt. Assuming it weighed about 5 kg, since the bolts took about 1 second to travel the roughly 500-foot distance, the momentum was approximately 760 kg*m/s. If it impacts over 0.0001 seconds and the tip has an area of 0.000001 m^2, then the maximum pressure at impact is 7600 GPa! That's insane!
However, the bolt got through, and anyway this calculation is maximum.
Drogon also took a blast of blue flame just fine for something like twenty seconds. This suggests that his heat tolerance is similar to Smaug's, or even a bit superior.
Strength:
Smaug's strength
Instances where Smaug showed immense strength can be found here. These include smashing through the iron-bound oak doors of Erebor along with the stone archway, ramming through a stone bridge, and breaking and pushed over a massive stone pillar, and he was only knocked over by a massive wave of liquid gold, which is more dense than lead.
Drogon's strength
The biggest strength feat I can find for Drogon is that he can fly. That's really impressive. Sure, so can Smaug, but that means that Drogon is proportionally about as strong as the firedrake.
He was also not overpowered by dozens of wights attacking at once.
Firepower:
Smaug's fire
Smaug can engulf huge areas in flame with a temperature over 1200 degrees. His blasts can have a total yield of over 10 kilotons, and can wipe out wooden and even stone structures. He blasted through the main arch of Erebor as well. The total energy can exceed 40TJ, and the effective energy (what is transferred to the target) is almost certainly in the hundreds of gigajoules.
Drogon's fire
Drogon turned armored men to ash (which requires at least 260 megajoules each)* with a few seconds of flame at most. He also melted a throne of solid iron with an extended stream, which is a gigajoule-level to terrajoule-level feat. He also boiled a few hundred gallons of water with a blast, killing two men who dove into it.
*Let's say the total energy of his blast is 100 times this. In that case his power output is several gigawatts. Not bad.
Conclusion:
Smaug is bigger, stronger, more intelligent, has a greater fire output, and is faster. Drogon is probably more agile, and that's it. Their armor is actually similarly strong, but even Smaug's wings have shown this level of anti-piercing durability, whereas Drogon's in my knowledge have not.
But still, it would be really cool to see. Drogon could put up a fight, and unlike Toothless he doesn't need to dodge to avoid being one-shotted by Smaug, being very durable.
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